Shape and topology optimization of conformal thermal control structures on free-form surfaces: A dimension reduction level set method (DR-LSM)

Abstract

In this paper, the authors propose a dimension reduction level set method (DR-LSM) for shape and topology optimization of heat conduction problems on general free-form surfaces utilizing the conformal geometry theory. The original heat conduction optimization problem defined on a free-form surface embedded in the 3D space can be equivalently transferred and solved on a 2D parameter domain utilizing the conformal invariance of the Laplace equation along with the extended level set method (X-LSM). Reducing the dimension can not only significantly reduce the computational cost of finite element analysis but also overcome the hurdles of dynamic boundary evolution on free-form surfaces. The equivalence of this dimension reduction method rests on the fact that the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar with the conformal mapping. The proposed method is applied to the design of conformal thermal control structures on free-form surfaces. Specifically, both the Hamilton–Jacobi equation and the heat equation, the two governing PDEs for boundary evolution and thermal conduction phenomena, are transformed from the manifold in 3D space to the 2D rectangular domain using conformal parameterization. The objective function, constraints, and the design velocity field are also computed equivalently with FEA on the 2D parameter domain with properly modified forms. The effectiveness and efficiency of the proposed method are systematically demonstrated through five numerical examples of heat conduction problems on the manifolds.